Molecular Dynamics Simulation of Homogeneous Crystal Nucleation

Article pubs.acs.org/Macromolecules

Molecular Dynamics Simulation of Homogeneous Crystal Nucleation in Polyethylene Peng Yi,† C. Rebecca Locker,§ and Gregory C. Rutledge*,‡ †

Department of Physics and ‡Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States § ExxonMobil Research and Engineering Company, Annandale, New Jersey 08801, United States ABSTRACT: Using a realistic united-atom force field, molecular dynamics simulations were performed to study homogeneous nucleation of the crystal phase at about 30% supercooling from the melts of n-pentacontahectane (C150) and a linear polyethylene (C1000), both of which are long enough to exhibit the chain folding that is characteristic of polymer crystallization. The nucleation rate was calculated and the critical nuclei were identified using a mean first-passage time analysis. The nucleation rate was found to be insensitive to the chain length in this range of molecular weight. The critical nucleus contains about 150 carbons on average and is significantly smaller than the radius of gyration of the chains, at this supercooling. A cylinder model was used to characterize the shape of the crystal nuclei and to calculate the interfacial free energies. A chain segment analysis was performed to characterize the topology of the crystal surface in terms of loops (including folds) and tails. The length distribution of loops is broad, supporting the “switchboard model” for the early stage crystals formed at deep supercooling. Using the survival probability method, the critical nucleus size was determined as a function of temperature. The interfacial free energies were found to be temperature-dependent. The free energy barrier and nucleation rate as functions of temperature were also calculated and compare favorably with experiments.

I. INTRODUCTION The crystallization of polymers, like that of small molecules, can be described by a two-stage process of homogeneous nucleation and crystal growth. Significant understanding of the second stage, crystal growth, has been achieved as a result of several decades of investigation, using a variety of techniques including optical microscopy, light scattering, X-ray scattering, atomic force microscopy, calorimetry, and molecular simulation; a number of texts and reviews are available.1−9 However, the initial stage involving nucleation of polymer crystals has received much less attention, despite early efforts in this direction over 50 years ago.10 Most of the previous studies of polyethylene that focused on homogeneous nucleation10−14 relied on the droplet technique,15 where a volume of melt is dispersed into a large number of droplets, N0. N0 is chosen to be large so that only a small fraction of droplets contain impurities, and the droplets are small (on the order of micrometers) so that each droplet can accommodate only one nucleation event. Crystallization of droplets is often detected using optical microscopy, and the fraction of crystallized droplets NC/N0, is recorded as a function of time. The theoretical form of NC/N0 is known for steady-state nucleation under classical nucleation theory, and can be used to fit the experimental data to estimate kinetic and thermodynamic properties.16 Many factors can affect the interpretation of the experimental results, including cooling rate, droplet sizes and their polydispersity, thermal history, effects of suspending © XXXX American Chemical Society

media or substrates, and validity of the steady-state nucleation assumption.17 As a result, given the relative paucity of experimental investigations, our understanding of the nucleation of polymer crystals on the microscopic level is sorely incomplete. For example, one very interesting phenomenon that is unique to polymer systems is chain folding during crystallization.18,19 Despite much experimental effort, it still remains unclear at what point during crystallization, and by what mechanism, the characteristic folded configuration of lamellar polymer crystallites is realized during the liquid-tosolid transition. Compared to crystallization from solution, this problem is particularly complicated in crystallization from the melt, because the nucleation process and the subsequent thickening process, during which crystal nuclei grow in the chain axis direction, are difficult to separate. Nevertheless, chain folding and development of the lamellar crystalline morphology is essential to the structure and properties of the majority of commodity and high performance plastics. Given the lack of spatial and temporal resolution of current experimental techniques, computer simulation offers a valuable alternative to improve our understanding regarding homogeneous nucleation. Furthermore, the study of homogeneous nucleation is particularly suited to molecular simulation Received: March 4, 2013 Revised: May 3, 2013

A

dx.doi.org/10.1021/ma4004659 | Macromolecules XXXX, XXX, XXX−XXX

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approximated by ΔGv ≈ ρnΔHfΔT/Tm, where ρn is the molecule number density, ΔHf is the heat of fusion per molecule at the thermodynamic equilibrium melting temperature Tm, and ΔT (equal to Tm − T) is the supercooling. For deeper supercooling, a more accurate approximation40 is

because, first, the most interesting features are manifested on the nanometer length scale, and second, unlike experiments, simulation permits full control over the composition of the system to ensure impurity-free homogeneous nucleation conditions. However, homogeneous nucleation is also a rare event, which tends to be particularly challenging for computeintensive molecular simulations, due to the potentially prohibitive waiting times between short bursts of interesting dynamics. To overcome this problem, it is common practice in molecular simulations of polymer crystallization to bias the simulation in a manner that reduces the waiting times or eliminates them altogether. Such approaches that have been used in the past include restriction to short chains,20−23 use of preoriented melts,24−27 artificially stiffening or linearizing the chain backbone,28,29 coarse graining of the polymer chain beyond the level of united atoms,30−33 or surface-induced preordering.34,35 In addition, high degrees of supercooling are normally employed to achieve nucleation rates high enough to observe nucleation in the much smaller volumes of simulation cells as compared to sample volumes used in experiments. However, a high degree of supercooling also results in higher melt viscosity, lower diffusivity, and longer relaxation times. Therefore, the temperature range within which the relaxation time is small enough and the nucleation rate is high enough to be observed by current MD simulations can be very narrow. To our knowledge, there does not exist any simulation work that fully characterizes homogeneous nucleation in a polymer melt, complete with determination of the nucleation rate and identification of the transition state. In this work, we study homogeneous crystal nucleation from the melt for the oligoethylene n-pentacontahectane (C150) and a short polyethylene (C1000). This is a continuation of our previous work for the normal alkanes n-octane (C8) and neicosane (C20).22,23 Our motivation for selection of C150 and C1000 for this study is two-fold. First, C150 is the shortest chain to display chain folded crystals (from solution).18 Second, the entanglement length of polyethylene is about 60−90 carbons determined by experiments36,37 and simulations;38,39 C150 is about twice this length, thereby ensuring that the effects of entanglements on crystallization kinetics are captured. Both folding and entanglement are essential characteristics of polymer crystallization from the melt. This paper is organized as follows. We first present the theoretical background and analytical method, followed by simulation details. Then we will discuss our simulation results and compare with existing experimental studies.

ΔGv ≈ ρn ΔHf T ΔT /Tm2

Maximizing ΔG with respect to r and l gives the critical radius r* and critical length l*, respectively. ⎧ r* = 2σs/ΔGv ⎨ ⎩ l* = 4σe/ΔGv ⎪ ⎪

(3)

Inserting the values of r* and l* into eq 1, we obtain the free energy barrier,

ΔG* = 8π

σs 2σe (ΔGv )2

(4)

The critical nucleus size, measured in number of molecules, is

n* = ρn π(r*)2 l* = 16πρn

σs2σe (ΔGv )3

=

2ρn ΔGv

ΔG*

(5)

According to transition state theory, the nucleation rate I (in number of events per unit volume per unit time) is given by

I = I0 exp(−ΔG* /kBT ) = A exp(− Ed /kBT ) exp(−ΔG* /kBT ) (6) where Ed is the activation energy for processes that transport chain segments to or from the nucleus, e.g., by diffusion, and A is a temperature insensitive prefactor. For steady-state nucleation, in which the size distribution of nuclei is time-independent despite the fact that there is a constant net flux of molecules from smaller nuclei to larger ones, the induction time τ* is the time elapsed before a critical nucleation event occurs. The nucleation rate can be calculated from I = 1/(τ*V), where V is the volume of the system.41 The term “induction time” or “incubation time” is often used in crystallization studies to denote the time elapsed before the establishment of steady-state nucleation in a bulk;4 it should not be confused with the induction time of classical nucleation theory, as employed here. A mean first-passage time (MFPT) analysis42 can be applied to the largest nucleus, nmax, observed in MD simulations in order to estimate the induction time. This implies that nmax is taken to be the reaction coordinate for the nucleation process.16 For a process with a sufficiently high barrier, ΔG* ≫ kBT, the MFPT for nmax follows the equation

τ(nmax ) = 0.5τ*[1 + erf(Z π (nmax − n*))] + G−1(nmax − n*)H(nmax − n*)

(7)

where n* is the critical nucleus size, Z is the Zeldovich factor, G is the growth rate, and H(x) is the Heaviside function. The second term on the right-hand side has been added to the original equation of ref 42 to account for finite crystal growth rates of postcritical nuclei, when G≫n*/τ* is not satisfied. The Heaviside function was used because growth is defined only after nucleation has occurred. An error function can be used for a smooth approximation of the Heaviside function, i.e.

II. THEORY AND METHOD Classical nucleation theory (CNT) has been widely used to describe homogeneous nucleation. The crystallites of chain molecules and polymers are necessarily anisotropic and exhibit at least two distinct types of surfaces: (i) chain-end or chain-folded surfaces, and (ii) lateral or side surfaces. For this reason, a cylinder model is often assumed to describe the shape of crystallites in polymer systems. Our recent studies of C8 and C20 confirm unambiguously that this is the simplest model that captures the shape of the free energy curve for homogeneous nucleation.22,23 The free energy of formation of a cylindrical crystal nucleus is given by

ΔG = 2πr 2σe + 2πrlσs − πr 2l ΔGv

(2)

τ(nmax ) = 0.5τ *[1 + erf(Z π (nmax − n*))] + 0.5G−1(nmax − n*)[1 + erf(C(nmax − n*))]

(8)

where C → ∞ is chosen to be a large positive number in practice. This method has been applied to MD simulations of homogeneous crystal nucleation in two shorter n-alkane systems22,23 at about 20% supercooling. MFPT analysis relies on an unbiased MD simulation of the reversible growth and shrinkage of the largest nucleus until such time as it crosses the free energy barrier and proceeds to grow irreversibly. At shallow supercooling, this is not practical because the free energy

(1)

where σe and σs are the interfacial free energies for the end and side surfaces of a cylindrical nucleus of radius r and length l, and ΔGv is the Gibbs free energy difference per unit volume between the crystal and melt phases. For a small degree of supercooling ΔT/Tm, ΔGv can be B


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where θij is the angle between the vector from the (i − 1)th UA to the (i + 1)th UA, and the vector from the (j − 1)th UA to the (j + 1)th UA. The average in eq 9 is taken over all of the j neighboring UAs of UA i that lie within a cutoff distance rij < rp2. By contrast, the global orientation order parameter P2 is defined as

barrier increases with temperature according to eqs 2 and 4, resulting in prohibitively long induction times for simulation studies close to Tm. Therefore, to calculate the critical nucleus size n* at higher temperatures, we used a survival probability method.43 According to CNT, the critical state, comprising the ensemble of configurations containing a critical nucleus, is a point of unstable equilibrium, with equal probability for a given configuration to evolve to either the fully crystallized or molten state. In the survival probability method, we prepare an ensemble of configurations in which the largest nucleus size is n0 and then run MD simulations on each configuration at different temperatures. There exists one temperature Tc at which the configurations in the ensemble exhibit a 50% probability to crystallize (or melt); n0 is then equated with the critical nucleus size at this Tc. This method has been previously applied with some success to the Ising model43 and the Lennard-Jones system.44,45

P2 =

j

i≠j

(10)

IV. RESULTS AND DISCUSSION a. Properties of the Amorphous and Crystal Phases. The glass transition temperature Tg was first estimated to identify the proper temperature range for crystallization simulation. An amorphous system was cooled at a finite cooling rate, and the specific volume as a function of temperature was monitored, as shown in Figure 1. The slope

Figure 1. Specific volume as a function of temperature during cooling simulations of a C150 melt at three different cooling rates, 100, 25, and 10 K/ns, respectively. (Inset) The glass transition temperature, Tg, as a function of cooling rate and the extrapolation of Tg to zero cooling rate.

of this curve is the thermal expansion coefficient. A change of the thermal expansion coefficient indicates the apparent glass transition. The apparent glass transition is known to be cooling rate-dependent, so that the high cooling rates used in simulations generally lead to a higher estimate of Tg compared to experimental methods. To account for this, three different cooling rates were used, 100, 25, and 10 K/ns. A plot of apparent Tg versus cooling rate was then extrapolated to zero cooling rate, as shown in the inset of Figure 1, resulting in the estimate Tg0 = 215.22 ± 1.24 K for C150. Following the same procedure, Tg0 was found to be 223.04 ± 0.22K for C1000, which is in good agreement with experimental values for

3 cos2 θij − 1 2

2

where the average is taken over all pairs of (i, j) in the system regardless of the distance between i and j. A threshold p2,th can be chosen so that UAs with p2 greater than p2,th are assigned to the crystal phase. Two crystal UAs i and j are assigned to the same crystal nucleus if rij is less than a threshold value rth. The size of a crystal nucleus is the number of UAs assigned to that nucleus. Consistent with our previous study of n-eicosane,23 the values of parameters, rp2, rth, and p2,th, are chosen to be 2.5σ, 1.3σ, and 0.4, where σ is the Lennard-Jones interaction range parameter.

III. SIMULATION DETAILS As in our previous studies of C8 and C20, we used the unitedatom force field originally proposed by Paul, Yoon and Smith (PYS)46 and subsequently modified by Waheed et al.47,48 This force field combines the hydrogens and carbon of each CH2 or CH3 moiety into a single united atom (UA) site, but otherwise retains all of the conformational degrees of freedom of a fully explicit atom model. It has been shown to reproduce a variety of melt phase structures and dynamical properties as well as the structure, melting point, and enthalpy of the rotator phase in short n-alkanes. It favors the hexagonal crystal phase of polyethylene over the more commonly observed orthorhombic crystal phase. Details of this force field may be found elsewhere.22 MD simulations were performed using the open source LAMMPS package.49 Isothermal−isobaric (NPT) ensembles were used with the pressure P = 1 atm. The equations of motion were integrated using the velocity Verlet method with an integration time step Δt = 5 fs. The details of the thermostat and barostat can be found in Yi and Rutledge.23 The simulations used to characterize the amorphous melts and to study crystal nucleation were performed on systems containing either 60 C150 chains or 9 C1000 chains. The simulation cells were orthorhombic with the side lengths changing independently. Periodic boundary conditions were applied in all three directions to approximate bulk-like behavior. The initial configurations were generated at a density of 0.3 g/ cm3 by randomly growing each chain with fixed bond lengths, fixed bond angles and torsion angles based on their energy distribution at 550 K, similar to the method used by Waheed et al.48 These configurations were equilibrated at 550 K for 100 ns before being quenched instantaneously to a lower temperature for simulation of the amorphous phase or studies of nucleation. The simulations used to characterize the crystal phase were performed on a system of 320 C150 chains. A triclinic simulation cell with all angles and side lengths varied independently was used, with periodic boundary conditions in all three directions. The initial configurations were generated using a hexagonal lattice with lattice constants a = 0.48 nm, and c = 19.5 nm. These configurations were equilibrated at 200 K for 20 ns before being heated instantaneously to a higher temperature to simulate the crystal phase properties. The crystal nucleus is defined based on a local p2 order parameter. The local p2 order parameter of the ith UA on a chain is defined as p2 (i) =

3 cos2 θij − 1

(9) C


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Figure 2. (a) End-to-end vector orientation autocorrelation function for C150 melt at 450 K. Simulation data are shown with open circles and linear fitting is shown with a straight line. (Inset) Rouse time as a function of inverse temperature for C150 melt. Line is provided as a guide to the eye. (b) Mean square displacements of center of mass of chains as a function of time for C150 melt at 450 K. (Inset) Diffusion coefficient of center of mass of chains as a function of inverse temperature, and linear fitting.

polyethylene, 120 K-220 K,.50 For comparison, a previous estimate of Tg using a smaller simulation of 4 C1000 chains, the same force field, a cooling rate of 125 K/ns and no extrapolation resulted in the estimate Tg(@125 K/ns)=280 ± 32 K.51 Other investigators also reported Tg between 200 and 250 K using simulation methods.52,53 In the following studies, we carried out crystal nucleation simulations at 280 K, which is well above Tg0. MD simulations were performed on C150 melts at different temperatures to calculate the end-to-end vector orientation autocorrelation function and the mean square displacement (MSD) of the centers of mass of the chains, as described in Harmandaris et al.54 These results are shown in Figure 2. From the slope of the logarithm of the end-to-end vector orientation autocorrelation function versus time, a characteristic relaxation time is obtained. This relaxation time is equated with the Rouse time τ R , invoking the assumption that hydrodynamic interactions are screened in the melt. The supercooled melt shows significant slowdown as the Rouse time τR increases much faster than the 1/T predicted by the Rouse model. From the MSD we calculate the self-diffusion coefficient, MSD=6Dtν, where ν is confirmed to be equal to unity at long time, according to Fick’s law.55 Our calculation shows that D = 4.1 × 10−7 cm2/s for C150 at 450 K and 1 atm, which agrees well with the empirical equation fitted from experimental data.56 This value is slightly lower than that reported by Harmandaris et al. using a different UA force field,55 D ≈ 7 × 10−7cm2/s for C156 at 450 K and 1 atm. The self-diffusion coefficients D obtained by simulations at several different temperatures in the range 280−450 K were fit to the equation ln D = ln D∞ − Ed/T where Ed is the activation energy for diffusion, as shown in the inset of Figure 2b. Our results show that Ed = 21.46 ± 3.30 kJ/ mol, which agrees well with experiments.56,57 In the experiments,56−58 Ed attains an asymptotic limit for long chains, e.g., N > 200 at T = 423 K.56 This suggests that Ed is a measure of local relaxation and that the molecular weight dependence of D is mainly controlled by the parameter D∞, i.e., D∞(N). Potential energy per chain and average density were extracted from simulations of the melt phase and of the crystal phase for C150 at each temperature. These data, shown in Figure 3, were extrapolated to the experimentally determined equilibrium melting point Tm = 396.4K18 for the calculation of

Figure 3. Temperature dependence of potential energy per chain (a) and average density (b), in the crystal phase and the melt phase in the simulated C150 systems. Dash lines indicate the melting temperature. Solid lines are provided as guides to the eye.

the heat of fusion ΔHf = ΔE + PΔV. From ΔE = 467.35 ± 17.15 kJ/mol and ΔV = 310.54 ± 20.06 cm3/mol, we obtain ΔHf ≈ 467 ± 17 kJ/mol of C150 molecules; this is equivalent to 53.1 ± 1.9 cal/g, which can be compared with the experimental value of 68.4 cal/g for polyethylene.59 As was noted previously for n-eicosane,23 the simulated value is predictably lower than the experimental value, since the PYS force field predicts crystallization into a rotator phase rather than the experimentally observed orthorhombic crystal phase of polyethylene. It is worth noting, however, that the heat of fusion for the rotator phase of n-eicosane, which is observed experimentally, was accurately reproduced by simulations using the PYS force field.23 b. Homogeneous Nucleation in C150 and C1000 Melts. Figure 4a shows plots of potential energy and nmax versus time from a representative MD trajectory of a C150 melt undergoing homogeneous crystal nucleation at 280 K. Three distinct time periods are observed: (1) re-equilibration of potential energy in the first ∼25 ns after quenching; (2) an induction period between 25 and 200 ns, during which time small fluctuations in both energy and nmax are observed; (3) nucleation and growth of a stable crystallite after ∼200 ns. D


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Figure 4. (a) Potential energy and the largest nucleus size, nmax, as functions of time during a typical MD trajectory of homogeneous crystal nucleation in C150 melt. The system was quenched from 550 to 280 K at time 0; (b) MFPT for C150 and C1000 at 280 K.

Table 1. Fitting Results of MFPT to Equation 8 for C150 and C1000 at 280 K system

Z (×10−3)

n* (UA)

τ* (ns)

I (1025 cm−3 s−1)

G (UA/ns)

C150 C1000

5.9 ± 0.8 5.9 ± 0.8

143 ± 14 170 ± 13

293 ± 19 340 ± 22

1.47 ± 0.10 1.26 ± 0.08

9.6 ± 2.7 4.5 ± 0.7

postcritical nucleus of size 800 UAs engages only relatively short segments of numerous C150 chains. In contrast to the nucleation rate, the growth rate differs considerably between C150 and C1000. Our observation of the molecular weight independence of nucleation rate is consistent with droplet experiments. Cormia et al.10 found that at deep supercooling the kinetic prefactor I0 was not significantly different between PE and n-alkane, therefore the chain segments involved in the diffusion processes must be similar in PE and in n-alkane. Massa et al.60 also found for poly(ethylene oxide) that the homogeneous nucleation rate is independent of molecular weight. Some experimental studies of polymer solution crystallization61,62 have been used to argue that nucleation rate is inversely proportional to molecular weight. However, it should be noted that these studies likely involved heterogeneous nucleation, and that the thermodynamic free energy barrier is nevertheless Mw-independent, since the nuclei are much smaller than the chain size. Thus, the nucleation prefactor must contain a Mw-dependent factor, which was attributed to the diffusivity D, in order to explain the experimental observations. These observations and our simulations can be rationalized if one realizes that the experimentally observable “nuclei” are much bigger than the critical nucleus size predicted by simulation. We posit that, due to limited spatial and temporal resolution of the methods, the experimentally observed nuclei are in fact postcritical crystallites, the development of which is sensitive to the molecular weight dependence of the growth process. While it is common to include the diffusivity D, or sometimes viscosity η, in the nucleation prefactor I0 for simple liquids, our simulations indicate that this cannot be the case for polyethylene, at least at 30% supercooling. Instead, the nucleation prefactor must depend on a segmental motion that is localized and not molecular weight dependent. The shape of the crystal nucleus is characterized using the cylinder model. The stem length, or thickness, of crystal nuclei l was calculated as a function of nucleus size n and is shown in Figure 6a. Fitting this data to a power law produces a scaling

Using independent starting configurations, 16 trajectories for C150, and 10 trajectories for C1000, were collected, from which the mean first-passage time of nmax was obtained for both C150 and C1000, as shown in Figure 4b. Equation 8 was fitted to each of these curves, as shown in Figure 4b, for the estimation of induction time τ*, the critical nucleus size n*, the Zeldovich factor Z, and the growth rate G. The parameter C in eq 8 was fixed to a large number during the fitting process. When C is greater than 100, the values of τ*, n*, Z, and G are invariant with C; they are reported in Table 1. Remarkably, we find that the nucleation rates are almost identical for C150 and C1000 at 30% supercooling, and are only about a factor of five slower than that obtained previously for C20 at 20% supercooling. This confirms that nucleation is a local event. At such deep supercooling, nucleation is not affected strongly by the molecular weight, but only by the local environment. The snapshot in Figure 5 shows that even a

Figure 5. Snapshot of a crystal nucleus of size 800 in a supercooled C150 melt. (a) Only chains that participate in the nucleus are shown. The UAs in the crystal nucleus are shown in red while the UAs that are in the melt phase are shown in blue. (b) Only UAs that are in the nucleus are shown (side view). (c) Only UAs that are in the nucleus are shown (top view). A cylinder shape illustrated by black dashed lines is provided as a guide for the eye. E


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Figure 6. (a) Thickness of cylindrical crystal nucleus as a function of nucleus size. (b) Number of tails and loops as functions of nucleus size. (c) Number of tails and loops as functions of their respective contour lengths, for nuclei of size about 800 UAs.

much greater than 2 even when nucleus size n < 100 UAs, suggesting that there are multiple chains participating in the crystal nucleus during the critical nucleation event, rather than a single chain as suggested by the “intramolecular nucleation” mechanism.64,65 The contour length distributions of tails and loops, Ntail(k) and Nloop(k), are also calculated for nuclei of size 800 ± 25 UA. It is clear from Figure 6c that both types of segments have broad distributions of length. This finding supports the “switchboard” re-entry model proposed by Flory and Yoon,66 at least for early stage crystallites at deep supercooling. The contour length distribution of both tails and loops shows an exponential decay as a function of contour length, suggesting a constant incremental potential, μUAsim = −kBT[∂(ln N)/∂k], for each UA on polymer chains. From the slopes of the curves in Figure 6c for k > 30, values of −0.013 and −0.077 kJ/mol may be estimated for the incremental chemical potentials of tails and loops, respectively. These values can be compared to μUAsim ∼ −0.033 kJ/mol obtained from simulations of melts of linear alkanes at P = 1.02 atm,67 μUAexpt ∼ −0.039 kJ/mol from fitting of experimental data using the Peng−Robinson equation of state68 and μUAsim ∼ −0.03 kJ/mol obtained from Monte Carlo simulation of interlamellar isotactic polypropylene.69 The difference between the values for incremental chemical potential of loops and tails estimated here, and the values determined from melts, presumably traces back to the relatively random nature by which segments of chains are incorporated kinetically into the nucleus.

coefficient of 0.31, which is close to 1/3 and suggests that small nuclei grow in three dimensions. Finding l* for n* at 280 K, we calculated the interfacial free energies to be σs = 21.4 mJ/m2 and σe = 12.4 mJ/m2 at 280 K. These values are higher than those previously calculated for C8 and C20,22,23 a trend that is consistent with experiments.63 The amorphous chain segments attached to the crystal nuclei tend to repel each other because of the lower density in the melt. This repulsion creates excess stress on the interface, which increases as the length of the amorphous segments increases. We also characterized topology of the crystal surface in terms of these amorphous chain segments, in particular loops and tails that belong to chains with segments incorporated into the nuclei. A loop is defined as an amorphous chain segment that connects two crystal stems in the same nucleus, and a tail is defined as an amorphous chain segment with one end attached to one crystal nucleus and the other end free in the melt. The term “fold” is reserved here for a loop with fewer than 10 carbons. As shown in Figure 6b, the number of tails and loops, Ntail and Nloop, scale with the size of the nucleus as Ntail ∼ n0.43 and Nloop ∼ n1.1, respectively. Since the number of UAs in the nucleus is proportional to the nucleus volume, the crosssectional area of the cylindrical nuclei, S, is proportional to n/l. Then according to Figure 6a, S is proportional to n0.69, and Ntail and Nloop are proportional to S0.62 and S1.6, respectively. For small nuclei, tails are more common than loops; as nuclei increase in size, the likelihood that two or more stems belong to the same chain increases, thereby accounting for the stronger dependence of Nloop on cylinder area S. It is clear that Ntail is F


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Figure 7. (a) MD simulations of C150 melts at different temperatures using the same initial configuration with nmax = 400 UA. (b) Probability of melting at different temperatures with an initial nmax = 400. (c) Critical nucleus size n* as a function of (TΔT)−1 for C150 melts. (d) Calculated critical thickness of nucleus as a function of supercooling; see text for details. (e) Calculated interfacial free energies as a function of temperature; see text for details.

c. Temperature Dependence of Nucleation Rate for C150. The survival probability method was used to determine the temperature dependence of the critical nucleus size. MD simulations were performed using ensembles of C150 systems with initial values of nmax = 150, 200, 400, 600, and 800 UAs, respectively. Each ensemble contained eight different initial configurations. The initial configurations were extracted from the previous MD simulations used to observe nucleation at 30% supercooling. Ensembles of configurations generated at such deep supercooling may differ in subtle ways from those relevant at shallower supercooling. When the temperature is changed,

degrees of freedom orthogonal to the reaction coordinate nmax (for example, the dimensions of the crystal unit cell within the nucleus) could change. However, so long as the re-equilibration times for these orthogonal degrees of freedom are short relative to the growth or shrinkage of the nucleus used to determine survival probability, the errors introduced by this method of initial structure generation are likely to be small. Figure 7a shows one set of simulations with an initial nmax = 400 at four different temperatures, 280 K, 290, 300, and 310 K. The system melted at 300 and 310 K, and crystallized at 280 and 290 K. From this one concludes that the critical nucleus size n* is G


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Figure 8. (a) Free energy barrier ΔG* as a function of temperature; (b) nucleation rate I as a function of temperature.

energies are temperature independent.45 Figure 7e shows the product of interfacial free energies σs2σe as a function of temperature calculated based on the data of Figure 7c, eqs 2 and 5. Furthermore, Figure 7d and eq 3 were used to calculate σe individually; σs and σe can then be separated, and are also shown in Figure 7e. This analysis indicates that the temperature dependence is much more significant in C150 than in our previous simulation with short n-alkanes, and nonmonotonic − the interfacial free energies take a maximum value at around 320 K. One possible explanation is that there are significant and nontrivial energetic and entropic contributions arising from the onset of chain folding and looping. Loops, and folds in particular, give rise to some crowding in the interface and a rise in the internal energy contributions to interfacial free energy.71 However, at still higher temperatures, the topological entropy due to the introduction of loops and folds becomes important, eventually offsetting the internal energy contributions so that the effective interfacial free energy decreases at high temperature. This behavior distinguishes the critical nuclei of C150 and polyethylene from the bundle-like nucleus previously reported for C20.23 Figure 8a shows the free energy barrier ΔG* calculated using classical nucleation theory (eqs 2 and 5) from the linear relation in Figure 7c, based on the values of n* obtained by survival probability analysis. To calculate the nucleation rate by eq 6 over the range of temperatures, we only require the values of Ed and A. Ed has already been calculated from data in Figure 2a and shown to be chain length independent. We note from Figure 8a that the free energy barrier ΔG* is about 40 kJ/mol at 280 K, but drops to 8 kJ/mol at 270 K, while the activation energy Ed = 21.46 kJ/mol. Therefore, we expect the kinetic contribution to become rate-limiting for temperatures less than about 270−280 K for C150. Given that Ed is Mw-independent for sufficiently long chains, this temperature might serve as a general lower limit of temperature for the application of CNT. The constant A can be determined by calibrating the nucleation rate to the simulation result at 280 K, which gives A = 4.3 × 1036 cm−3 s−1. The kinetic prefactor I0 = A exp(−Ed/kBT) is then calculated to be at 4.2 × 1032 cm−3 s−1 at 280 K. Combining the result of ΔG*, Ed, and A, the nucleation rate as a function of temperature is calculated as shown in Figure 8b. In contrast to the case for simple liquids, the growing/ shrinking of crystallites during the nucleation process does not require the diffusion of the entire chains, particularly at deep supercooling. The nucleation prefactor I0 therefore depends

bigger than 400 at temperatures greater than or equal to 300 K, and smaller than 400 at temperatures less than or equal to 290 K. If this trend holds, at some temperature Tc between 290 and 300 K we should have n* equal to 400. For each ensemble, the probability of melting was fitted to an error function to estimate Tc, as illustrated in Figure 7b. In this way, we determined the critical nucleus size n* as a function of temperature; this is plotted in Figure 7c. As shown in Figure 7c, n* as a function of temperature is well described by an equation of the form n* = a + b/TΔT; the fitting parameters for C150 are determined to be a = −2164 ± 129 UAs and b = (7.43 ± 0.37) × 107 UAs·K2. Assuming that the functional dependence expressed by Figure 6a is invariant with temperature and using Figure 7c, the thickness, l*, of the critical nucleus can be estimated for C150 at any temperature, as shown in Figure 7d. This l* is approximately linear with 1/ ΔT at small supercooling. However, it should not be mistaken for the lamellar thickness often discussed in the context of crystal growth, which corresponds to the fastest lateral growth rate and has also been shown to be proportional to 1/ΔT, or to 1/TΔT if better approximation is desired at deep supercooling (eq 2). Here the critical thickness l* might be related to the initial lamellar thickness discussed by Barham et al.;70 it indicates at what temperature crystallites with a certain thickness can form. For example, equating l* from Figure 7d with half the contour length of the fully extended C150 chain, one can obtain an estimate of ΔT = 2K for the supercooling at which a once-folded critical nucleus of C150 would be observed. On the basis of this estimate, any supercooling deeper than 2 K should result in once- or multiply-folded crystallites, yet folded C150 crystallites are not observed in melt crystallization experiments. That is probably because the isothermal thickening rate is fast on the time scale of observation. As a rough estimate, assuming that the crystal nucleus size n increases linearly with time at the growth rate G estimated in the foregoing MFPT analysis, then the time needed for a crystal nucleus of C150 chains to reach the extended chain thickness is on the order of 10−4 s, while that for C1000 chains is on the order of 10−1 s. Thus, as the chain length increases, folded chain crystallites become kinetically stabilized and are thus more likely to be observed. According to Figure 7c, n* is found empirically to vary linearly with (TΔT)−1. This contrasts with the linear dependence on (TΔT)−3 observed for crystallization of the simple Lennard-Jones system, where the interfacial free H


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estimate of I0 than the interfacial free energies from experiments. Fast scanning DSC methods currently under development may offer an alternative means for measuring nucleation rates, but results for polyethylene are not yet available.77 Crystallization of block copolymers could also provide an alternative to the classical droplet technique for studies of homogeneous nucleation.78

only on a segmental diffusivity Dseg. However, the temperature dependence, i.e., exp(−Ed/kBT), remains in effect as the temperature changes the rate of diffusion. Other temperature insensitive factors are relegated to the factor A. We postulate that this characteristic segment length is limited by the mesh size created between entanglements, and the relevant dynamics scale with the Rouse dynamics of chain segments of entanglement length.72,73 Further investigation is required to identify the characteristic segment length as a function of temperature during polymer nucleation, and thereby provide a physical interpretation for A. d. Comparison to Experiments. The experimental results for homogeneous crystal nucleation from the melt are rare for long n-alkanes or polyethylene; a number of these are summarized in Table 2. The experimental measurements of

V. CONCLUSIONS We have demonstrated that it is feasible to observe homogeneous nucleation of the crystal phase from the polyethylene melt at deep supercooling using molecular dynamics. This is attributed to the finding that the crystal nuclei are small relative to the radii of gyration of the chains, suggesting that polymer crystal nucleation is a local event. The nucleation behavior (dynamics and structure) itself is insensitive to chain length under these conditions. The growth rate, on the other hand, clearly decreases as the chain length increases. On the basis of this chain length independence, we postulated that the important unit during polymer crystal nucleation is a subchain segment, the length of which is selected by temperature and possibly entanglements; when the thickness of the nucleus exceeds the mesh size of the entanglement network, extrapolations such as the one shown in Figure 8 are likely to become inaccurate. We characterized the shape of crystal nuclei using a cylinder model. For the deep supercooling employed here, the initial thickness of the critical nucleus is much shorter than the chain length, and the crystal stems are contributed by multiple chains, in contrast to the “intramolecular nucleation” mechanism. Amorphous chain segments form tails and loops that are attached to crystal nuclei. Both the tails and the loops have broad length distributions and suggest that the UAs on the chains have constant incremental potential. The survival probability method was used to study critical nuclei over a broad range of temperatures. The thickness of the critical nucleus decreases with increasing supercooling, which might account for the onset of chain folding and looping with decreasing crystallization temperature or increasing molecular weight. The interfacial free energies were calculated using a cylinder model and found to be temperature dependent, with a maximum at about 320 K. Although we observed that the crystal growth process is 3-dimensional immediately beyond the critical nucleation event, the ultimate shape of the crystal lamella is still expected to result from a competition between the thickening rate and the lateral growth rate, which diverge from each other as the crystallite grows. Within the framework of classical nucleation theory, we have calculated the free energy barrier and its dependence on temperature. The diffusive and energetic contributions to the nucleation rate could thus be separately evaluated. By so doing, this work provides the first quantitative model of homogeneous nucleation of a polymer melt from first principles; it exhibits satisfactory agreement with existing experimental results. The simulations and theoretical analysis performed in this work constitute a set of methodologies that serve not only to reveal the molecular mechanisms underlying homogeneous crystal nucleation but also generate meaningful rate equations that can be used for engineering design purposes. It is our hope that the results presented here prove to be both enlightening and of practical utility to the polymer crystallization community.

Table 2. Experimental Estimates of Nucleation Kinetic Prefactor, I0, and the Product of Interfacial Free Energies, σs2σe, as Compared with our Simulation Result reference Cormia et al.10 Gornick et al.11 Koutsky et al.12 Ross and Frolen13 Kraack et al.14 this work a

system

Tc (K)

ΔT (K)

PE PE PE PE

360 362 360 358

55 53 55 57

∼C140 C150

349 280

47.6 116.4

I0 (cm−3 s−1) 1030.1 1047 2 × 1025 1049.4 1038 4.2 × 1032

σs2σe (mJ3/m6) 15500 14960 6800 19000 6349 5636a

See text for different values at other temperatures.

σs2σe, the product of interfacial free energies, vary by a factor of 2 to 3. Our simulation results are of the same order of magnitude, with σs2σe = 5636 mJ3/m6 at 280 K, 15435 mJ3/m6 at 340 K and 9880 mJ3/m6 at 360 K. It is not generally possible to obtain values of σs and σe individually from nucleation experiments. A second experiment is required. One common approach is to measure the initial lamellae thickness as a function of supercooling to derive the value of σe based on a kinetic theory.2,70 A second approach is to measure the melting temperature of lamellae as a function of lamellar thickness to derive the value of σe, based on the Gibbs−Thomson equation.2 Both methods have yielded values of σe∼90 mJ/m2, which is significantly larger than our result of σe = 12.4 mJ/m2 at T = 280 K. We believe the difference is attributable to the mature nature of the lamellae used in these latter experiments, in contrast to the nm-sized nuclei relevant during nucleation. This difference was also observed by Okada et al.,74 who measured the nucleus size distribution by SAXS technique and reported σe =18.5 mJ/m2, in closer accord with our value of 12.4 mJ/m2. Okada et al. hypothesized that σe increases to ∼90 mJ/m2 as the crystal lamellae grow, a process during which the surface topology changes. The determination of σs from experiments is even less straightforward and requires certain approximations. The value of σs calculated from the characteristic ratio C∞ is in the range of 10−20 mJ/m2.75,76 Our value of σs = 21.4 mJ/m2 at T = 280 K is slightly higher than this range. Our simulation results of I0 also fall within the range of experimental data, although it should be noted that the range of experimental measurements is quite large. The experiments are known to be extremely sensitive to any residual ordering that survived the melting process. It is more difficult to get a reliable I


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AUTHOR INFORMATION

Corresponding Author

*(G.C.R.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant Number OCI-1053575. We appreciate many helpful discussions with Dr. Andy Tsou. Financial support from Exxon-Mobil is gratefully acknowledged.

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Molecular Dynamics Simulation of Homogeneous Crystal Nucleation

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